MOSers at the hub of the next zig or zag

This post attempts to outline a method of generating scales (or sets
and subsets) that is itself a sort of derailed offshoot of an attempt
to recast, soften up, or perhaps 'un-temper' some of the more
'difficult' low order equal divisions of the octave (8 & 11 for
instance)... It was inspired in part by the following TD exchange
between Paul Erlich and Kraig Grady:

[Paul Erlich:]
> In 19-tET, the 11-tone chain-of-minor-thirds MOS is 0 1 2 5 6 7 10
11 12 15 16 (19). It's improper because the largest "second" (3/19
oct.: 2 to 5, 7 to 10, 12 to 15, 16 to 19) is larger than the smallest
"third" (2/19 oct.: 0 to 2, 5 to 7, 10 to 12).

[Kraig Grady:]
>Thanks for clearing this up for me! Us MOSers tend to be attracted to
those points right before the hub of the next zig or zag. So in 19 et
we would gravitate to the 15 tone subset.

The use of this particular method grew out of my curiosity upon
discovering the near Pythagorean intonation derived by taking the mean
of twelve tone equal temperament, its fifth, and its fourth, i.e.,
12e, 7e, and 5e taken {[(12/12)*0...12]+[(7/12)*0...12]+[(5/12
*0...12]}/3. Trivially this would be a 12-note subset of 1260e, i.e.,
(12*7*5)*3. But perhaps not so 'trivial' is that this is a very simple
procedure where (generically speaking) the mean of x, y, and x-y,
produces a fifth that differs from 2:3 by only ~.050� (in other
comparative Pythagorean terms, the limma is approximated @ +~.25�, the
apotome @ -~.35�, and the Pythagorean comma @ -~.60�). Observing the
closeness of these approximations led me to experiment with both other
(non-twelve) equal divisions of the octave (x), and other (non-fifth)
generators (y, and therefore x-y).

Using the recently discussed example of Dave Keenan's 11-tone
chain-of-minor-thirds scale (in 19e this would be +1, +1, +3, +1, +1,
+3, +1, +1, +3, +1, +3) as an example, you could derive the (near)
mode of the 5th degree of this scale when x=11, y=15 and x-y=-4, this
would be 0, 63, 253, 316, 379, 568, 632, 821, 884, 947, 1137, 1200.
Much as the 2nd degree near-Pythagorean scale of x=7 & y=12 is a (2nd
degree dorian*) subset of the original example of x=12 & y=7, the
11-note x=11 & y=15 could be said to be a subset of x=15 & y=11. This
is a mapping of 11 in one cycle (I'm considering a cycle to be x/2
through x*2).

y=5.5 0 109 218 327 436 545 655 764 873 982 1091 1200
6 0 103 219 322 439 542 658 761 878 981 1097 1200
7 0 94 230 323 417 553 647 783 877 970 1106 1200
8 0 86 256 342 429 515 685 771 858 944 1114 1200
9 0 81 162 398 479 560 640 721 802 1038 1119 1200
10 0 76 153 229 305 382 818 895 971 1047 1124 1200
12 0 70 139 209 279 348 852 921 991 1061 1130 1200
13 0 67 134 432 499 566 634 701 768 1066 1133 1200
14 0 65 292 357 422 487 713 778 843 908 1135 1200
15 0 63 253 316 379 568 632 821 884 947 1137 1200
16 0 61 228 289 455 517 683 745 911 972 1139 1200
17 0 150 210 360 420 570 630 780 840 990 1050 1200
18 0 138 197 334 472 531 669 728 866 1003 1062 1200
19 0 128 186 314 443 571 629 757 886 1014 1072 1200
20 0 121 242 298 419 540 660 781 902 958 1079 1200
21 0 114 229 343 458 572 628 742 857 971 1086 1200
22 0 109 218 327 436 545 655 764 873 982 1091 1200

When y=x/2, there is only one step size, and consequently s=L (I'm
using s and L here to generically denote small and large step sizes).
This example shows one cycle of step sizes where x=7 (excluding the 7e
sets where s=L & L=s):

4 s=4 & L=3 @: 0 157 348 505 695 852 1043 1200
5 s=5 & L=2 @: 0 137 394 531 669 806 1063 1200
6 s=6 & L=1 @: 0 124 248 371 829 952 1076 1200
8 s=6 & L=1 @: 0 107 214 321 879 986 1093 1200
9 s=5 & L=2 @: 0 102 448 549 651 752 1098 1200
10 s=4 & L=3 @: 0 97 368 465 735 832 1103 1200
11 s=3 & L=4 @: 0 230 323 553 647 877 970 1200
12 s=2 & L=5 @: 0 204 294 498 702 906 996 1200
13 s=1 & L=6 @: 0 185 371 556 644 829 1015 1200

Dan
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*As all of these scales are working from an equidistant division of
the octave back towards that equidistant division of the octave, they
are all, in a manner of speaking, dorian-esque mirror inversions.